Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 136750 by mnjuly1970 last updated on 25/Mar/21

           .....advanced    calculus.....       prove  that ::       ...  𝛗=∫_0 ^( ∞) ((1−e^(−x^2 ) )/x^2 )dx=(√π)

.....advancedcalculus.....provethat::...ϕ=01ex2x2dx=π

Answered by Ñï= last updated on 25/Mar/21

∫_0 ^∞ ((1−e^(−x^2 ) )/x^2 )dx=∫_0 ^∞ ((e^(−0) −e^(−x^2 ) )/x^2 )dx=∫_0 ^∞ ∫_0 ^1 e^(−ax^2 ) dadx  =∫_0 ^1 ((√π)/(2(√a)))da=(√π)

01ex2x2dx=0e0ex2x2dx=001eax2dadx=01π2ada=π

Commented by mnjuly1970 last updated on 25/Mar/21

   thanks alot...

thanksalot...

Answered by Dwaipayan Shikari last updated on 25/Mar/21

η(φ)=∫_0 ^∞ ((e^(−φx^2 ) −e^(−x^2 ) )/x^2 )dx⇒η′(φ)=−∫_0 ^∞ e^(−φx^2 ) dx=−((√π)/(2(√φ)))  η(φ)=−(√(πφ))+C⇒η(1)=0⇒C=(√π)   η(φ)=(√π) (1−(√φ)) ⇒η(0)=(√π)

η(ϕ)=0eϕx2ex2x2dxη(ϕ)=0eϕx2dx=π2ϕη(ϕ)=πϕ+Cη(1)=0C=πη(ϕ)=π(1ϕ)η(0)=π

Answered by mathmax by abdo last updated on 25/Mar/21

let f(a) =∫_0 ^∞  ((e^(−ax^2 ) −e^(−x^2 ) )/x^2 )dx  with a>0 ⇒f^′ (a)=−∫_0 ^∞  e^(−ax^2 ) dx  =−∫_0 ^∞  e^(−((√a)x)^2 ) dx =_((√a)x=t) −  ∫_0 ^∞  e^(−t^2 ) (dt/( (√a))) =−(1/( (√a))).((√π)/2) ⇒f(a)=(√π)∫ (da/(2(√a))) +C  =−(√(πa)) +C  f(1)=0=−(√π) +C ⇒C=(√π) ⇒f(a)=(√π)−(√(πa))  Φ=f(0)=(√π)

letf(a)=0eax2ex2x2dxwitha>0f(a)=0eax2dx=0e(ax)2dx=ax=t0et2dta=1a.π2f(a)=πda2a+C=πa+Cf(1)=0=π+CC=πf(a)=ππaΦ=f(0)=π

Terms of Service

Privacy Policy

Contact: info@tinkutara.com